L(s) = 1 | + (−2 + 2i)2-s + 5.34i·3-s − 8i·4-s + (10.1 + 10.1i)5-s + (−10.6 − 10.6i)6-s − 94.0·7-s + (16 + 16i)8-s + 52.4·9-s − 40.7·10-s + 104. i·11-s + 42.7·12-s + (−156. − 156. i)13-s + (188. − 188. i)14-s + (−54.4 + 54.4i)15-s − 64·16-s + (−288. − 288. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + 0.593i·3-s − 0.5i·4-s + (0.407 + 0.407i)5-s + (−0.296 − 0.296i)6-s − 1.92·7-s + (0.250 + 0.250i)8-s + 0.647·9-s − 0.407·10-s + 0.865i·11-s + 0.296·12-s + (−0.928 − 0.928i)13-s + (0.960 − 0.960i)14-s + (−0.241 + 0.241i)15-s − 0.250·16-s + (−0.998 − 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0764215 - 0.192023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0764215 - 0.192023i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 - 2i)T \) |
| 37 | \( 1 + (913. - 1.01e3i)T \) |
good | 3 | \( 1 - 5.34iT - 81T^{2} \) |
| 5 | \( 1 + (-10.1 - 10.1i)T + 625iT^{2} \) |
| 7 | \( 1 + 94.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 104. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (156. + 156. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (288. + 288. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + (257. + 257. i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 + (-226. - 226. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (59.2 - 59.2i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (608. - 608. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 - 1.81e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.88e3 - 1.88e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.51e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 185.T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.98e3 + 1.98e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (4.80e3 - 4.80e3i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 - 2.62e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.72e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 7.87e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.21e3 - 3.21e3i)T + 3.89e7iT^{2} \) |
| 83 | \( 1 + 9.87e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-3.65e3 + 3.65e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (-945. - 945. i)T + 8.85e7iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.93709066167101975934656503969, −13.36140408992317192074290145917, −12.57138741270628830747664221390, −10.62540770039526930089465118916, −9.802015921557124814342972162483, −9.284251286989555447111480406905, −7.21821240842939001821877723241, −6.48875639461906434805582356716, −4.77377963994803601405298060485, −2.82541742880188423561467186368,
0.11703406399883178732861041224, 2.06099650173271153187448753641, 3.84942072193650612333145556537, 6.17676702272708505280835142726, 7.12844157435768736294782743162, 8.848047337361044631904234483748, 9.646385439106593560833623169438, 10.74944302718198205043127720292, 12.46731075110566770426760302143, 12.80153914295967562438755122347